Shape-constrained nonparametric estimation of the term structure of interest rates

Abstract

This paper studies nonparametric estimation of the discount curve, which should be decreasing and positive over the entire maturity domain. Very few papers explicitly impose these shape requirements for removing the possibility of obtaining a shape-violating estimation. No matter how small the approximating error is, a shape-violating discount curve can never be accepted by the financial industry. Since these shape requirements are continuously constrained and involve an infinite number of inequality constraints, it is hard to provide a necessary and sufficient implementation that is computationally tractable. Existing parametric and nonparametric methods fail to achieve universal flexibility and shape compliance simultaneously. This paper proposes a nonparametric method that approximates the discount curve with algebraic polynomials and ensures the discount function is decreasing and positive over the entire domain. This estimation problem can be reformulated equivalently as a semidefinite program that is convex and computationally tractable. The proposed method is the first one which not only has asymptotic universal fitting flexibility, but also fully complies with shape requirements. Experimental results on one artificial data, one US Gilt STRIPS data, and one US Treasury bonds data demonstrate its superiority over state-of-the-art methods in terms of both the compliance of shape requirements and out-of-sample fitting measures.